5740
J. Phys. Chem. B 1997, 101, 5740-5746
Phenomenological Equations with Observable Electric Potentials Applied to Nonequilibrium Binary Electrolyte Solutions Javier Garrido* Departamento de Termodina´ mica, UniVersitat de Vale` ncia, 46100 Burjassot, Valencia, Spain
Vicente Compan˜ and Marı´a Lido´ n Lo´ pez Departamento de Ciencias Experimentales, UniVersitat Jaume I, 12071 Castello´ n, Spain
Donald G. Miller Geosciences and EnVironmental Technologies, Lawrence LiVermore National Laboratory, LiVermore, California 94550 ReceiVed: NoVember 25, 1996X
The irreversible processes which occur in a binary electrolyte solution are described by a new formalism which uses observable variables. This is based on (i) the relationship of the electrochemical potential of ion constituents to the chemical potential of the electrolyte; (ii) the notion of observable electric potential measured with reversible electrodes; and (iii) the thermodynamics of irreversible processes. The electric potential, the electric charge density, and the Nernst-Planck-Poisson formalism are reviewed in this work. The properties of the new formalism are evidenced by applying it to MgCl2-H2O solutions.
Introduction The difficulties which practical electrochemistry finds in the study of nonequilibrium systems are well-known. Most formalisms use nonobservable electric potentials, so that extrathermodynamic assumptions are needed if one wants to relate the theory to experimental measurements. However, new ways to study these systems have been developing for several years.1-9 The way described here is based on the observable electric potential (OEP).5-9 The value of this quantity at a point is the electric potential measured between a reversible electrode and another identical “reference” electrode always placed at a fixed point. The basic framework of the new formalism is the thermodynamics of irreversible processes. The phenomenological equations are deduced from the dissipation function in order to apply the Onsager reciprocal relations (ORR).10 The basic quantities of a formalism are the forces and fluxes. Here the thermodynamic forces consist of the gradients of all the electrolyte chemical potentials plus the gradient of an observable electric potential. The set of thermodynamic fluxes is formed by all but one of the ion constituent fluxes, plus the electric current density. Consequently this new formalism, all in terms of measurable quantities, helps complete the understanding of electrochemical systems provided by other formalisms. The OEP is directly related to the electrochemical potential difference of the ion for which the two electrodes are reversible. In this theoretical sense, the OEP is not a new concept. However, it is new in the sense of an alternative to the other constructs used in irreversible thermodynamics because it is a quantity that is measured experimentally. Consequently it provides a formalism for transport properties which is independent of unmeasurable quantities. A general theory of nonequilibrium electrolyte solutions using the OEP and the thermodynamics of irreversible processes has X
Abstract published in AdVance ACS Abstracts, July 1, 1997.
S1089-5647(96)03914-4 CCC: $14.00
not yet been fully achieved. Likewise a general statement about the advantages of the new formalism has not been presented. Until now it has only been applied to several special cases.5-9 One special case investigated was a ternary electrolyte solution with a common ion at uniform pressure and temperature.5 There the theory was used to evaluate the OEP from both experimental data and estimates of data. However, there are insufficient data to compare the OEP measurements with experiment and thus provide a proper illustration of their use. In fact no direct OEP measurements are available for comparison in any ternary system for which all the other data are available. Moreover, an OEP in three or more component systems is complicated by the fact that it depends on the concentration distribution of the electrolytes within the solution in the cell. In this paper, we provide in some detail the derivations and application of the new formalism to the simplest system, a binary electrolyte in a neutral solvent. The system selected for illustration, MgCl2-H2O, was chosen precisely because all the appropriate experimental data are available, including emf measurements which give one of the OEPs directly. In a binary system, an OEP is independent of the concentration distribution. We will also show that a binary system has two OEPs, one for each ion constituent. Each OEP is obtained from electrodes reversible to its corresponding ion constituent. As we will see later, this formalism is based on (i) the relationship of the electrochemical potential of ion constituents to the chemical potential of the electrolyte; (ii) the notion of observable electric potential measured with reversible electrodes; and (iii) the thermodynamics of irreversible processes. The classical decomposition of the electrochemical potential in two parts, one of chemical character and the other of electric properties, is not used here. The phenomenological equations are deduced from the dissipation function for nonequilibrium solutions and not from Maxwell’s equations of electrodynamics. Consequently we find it unnecessary to use Poisson’s equation, which relates the electric charge density to the electrical potential. © 1997 American Chemical Society
Nonequilibrium Binary Electrolyte Solutions
J. Phys. Chem. B, Vol. 101, No. 29, 1997 5741
Observable Electric Potential Formalism
X T Xz1 + z1e-
Consider a system containing a solution consisting of a binary electrolyte and an unionized solvent, denoted by the subscripts s and w, respectively. Therefore, there are only two ion constituents, denoted by the subscripts 1 and 2, whose charge numbers, of opposite sign, are z1 and z2, respectively. The stoichiometric coefficients ν1 and ν2 and the charge numbers are related by
ν1z1 + ν2z2 ) 0
(1)
One basic feature of the new formalism is the relationship between the electrochemical potential of ion constituents and the chemical potential of the electrolyte. Let ci be the concentration of ion constituent i (i ) 1, 2) in moles per liter, and let µ˜ i (i ) 1, 2) be the electrochemical potential of ion constituent i. When we refer these quantities to the concentration cs and to the chemical potential µs of the electrolyte, a difficulty arises. If the electroneutrality condition does not hold, the binary electrolyte is not rigorously defined at an arbitrary point in the solution. Therefore we cannot speak about a concentration and a chemical potential of the solute. This problem can be solved if we consider the following three system schemes. Let (cs)i ) ci/νi (i ) 1, 2) be the two values for cs, and let δcj ) cj - νj(cs)i (i * j ) 1, 2) be the deviation with respect to electroneutrality of the ion constituent j at a given point in the solution. At any such point, the same solution can be described by either (a) solvent plus both ion constituents i (i ) 1, 2), (b) solvent plus solute plus deviation of ion constituent 2, or (c) solvent plus solute plus deviation of ion constituent 1; i.e.,
(cwc1c2)
(2a)
(cw(cs)1δc2)
(2b)
(cw(cs)2δc1)
(2c)
Fortunately it is easy to demonstrate that the deviations δci, are extremely small, so that the two values (cs)i (i ) 1, 2) are essentially indistinguishable from a chemical point of view.11 Then if we apply Euler’s equation to these system schemes, we obtain
ν1µ˜ 1 + ν2µ˜ 2 ) µs
(3)
where µs is effectively a unique value of the chemical potential for the nearly identical concentrations (cs)1 ≈ (cs)2 defined above. The chemical potential µs is measured through the activity coefficient γs
µs ) µs* + νRT ln csγs
where e- denotes the electron. If we assume that (i) the concentration of electrode constituents X cannot vary and (ii) the pressure and temperature are uniform, then the OEP at any point will be
ψ1 )
1 ∆µ˜ 1 z1F
(6)
where as above we have taken the value of the OEP at the reference point as zero; F is Faraday’s constant; and ∆µ˜ 1 is the difference in the electrochemical potential between the two points of the solution where the electrodes are placed. Note that, under these assumptions, a change in the chemical nature of the electrodes X does not modify the measured value of these OEPs. For example, if both X electrodes are Cu (reversible to Cu2+), the measured potential is the same as if both X electrodes were Cu(Hg), both at the same amalgam composition. The “electric potential” of the solution will still be unknown, but the measured potential between the metallic phases of each pair of reversible electrodes will always give us a unique value, which will be called the OEP of Xz1. Since the binary electrolyte solutions have two ion constituents, there must be two OEPs, one for each. Thus the second OEP is measured with electrodes reversible to the other ion species Yz2. The OEP associated with Yz2 will be different from that for Xz1 but will be independent of the chemical nature of the Y electrode pair. For the gradients of the two OEPs we deduce in a general form
∇µ˜ i ) ziF∇ψi
i ) 1, 2
(7)
This result shows the simplicity of the formalism: the gradients in the OEP are the gradients in the electrochemical potentials divided by the constant quantity ziF and can be measured with any kind of electrodes reversible to species i (i ) 1, 2). We must remember that the system is at uniform pressure and temperature; under other conditions eq 7 has more terms. From eq 3 we obtain a relationship between the two OEPs:
∇ψ1 - ∇ψ2 )
1 ∇µs ν1z1F
(8)
The third part of the new formalism is the irreversible thermodynamics. If thermal equilibrium and uniform pressure are assumed, the dissipation function Tσ in our system is given by10,12 2
Tσ ) -
(4)
where ν ) ν1 + ν2 and µs* is the standard state chemical potential. The second basic feature of the new formalism is the concept of OEP. This quantity is the electric potential measured with two identical electrodes. These identical electrodes are reversible to one of the two ion constituents of the binary system. One is placed in a certain reference point, always the same, for which we can by convention assign the OEP as zero. The other electrode is placed in any arbitrary point of the solution. The OEP is the value of the electric potential measured potentiometrically between these two electrodes. Let the electrodes X for the chosen ion constituent be in equilibrium with the ionic species Xz1 of the solution; i.e.,
(5)
ji∇µ˜ i ∑ i)1
(9)
where ji is the flow density of matter of the ion constituent i in a solvent-fixed frame of reference. The local nonequilibrium state in the solution is described by the pair of driving forces (∇µ˜ 1, ∇µ˜ 2); but there are other ways to express the irreversibility. From eqs 3 and 7 we can obtain
∇µ˜ 2 ) z2F∇ψ2 ∇µ˜ 1 )
1 ∇µ + z1F∇ψ2 ν1 s
(10) (11)
and therefore we see that (∇µs, ∇ψ2) is an equivalent pair of driving forces. The fluxes conjugate to the new forces (j1/ν1, i)
5742 J. Phys. Chem. B, Vol. 101, No. 29, 1997
Garrido et al.
are deduced from the dissipation function, which can now be written as
1 Tσ ) - j1∇µs - i∇ψ2 ν1
These are related through
(12)
where i ) F(z1j1 + z2j2) is the electric current density. The phenomenological equations become
1 j ) -λ11∇µs - λ1ψ∇ψ2 ν1 1
(13)
i ) -λψ1∇µs - λψψ∇ψ2
(14)
X ) X(j)M(j)
j ) 1, 2
(27)
J(j) ) J(M(j))T
j ) 1, 2
(28)
where
M(1) )
[M(1)]-1 )
When we use the OEP ψ1, the analogous equations are
1 j ∇µ - i∇ψ1 ν2 2 s
Tσ ) -
(16)
i ) -λψ2∇µs - λψψ∇ψ1
(17)
with forces (∇µs, ∇ψ1) and fluxes (j2/ν2, i). The phenomenological coefficients λij (i, j ) 1, 2, ψ) are related to the three following experimental transport quantities: the specific electric conductance κ, the Hittorf transference numbers ti, and the diffusion coefficient D in the solvent-fixed frame12. From the phenomenological equations we deduce5
λiψ ) λψi )
(18)
tiκ νiziF
i ) 1, 2
(19)
[M(2)]-1 )
i ) 1, 2
J(j) ) -X(j)(λ(j))T
(32)
(33) j ) 1, 2
(34)
j ) 1, 2
(35)
1 lii νi2
i ) 1, 2
(36)
z1li1 + z2li2 F νi
i ) 1, 2
(37)
are related by
λ(j) ) M(j)L(M(j))T
λii )
λiψ ) λψi )
2
h
i ) 1, 2
i ) 1, 2
(21)
(22)
It is of course possible to relate the λik based on the OEP to the customary ionic Onsager coefficients10,12 based on ionic fluxes and on forces consisting of their corresponding electrochemical potentials ∇µ˜ i. In a matrix notation, the new and customary older10,12 forces and fluxes can be written
X(j) ) (∇µs, ∇ψj) J(j) )
( ) ν1 0 1 ν2 z F 2
J ) -XLT
(20)
Here we have assumed that ORR hold. If an ORR test is considered,12 then the phenomenological coefficients will be given in terms of emf transference numbers tic and Hittorf transference numbers tih
ticκ λψi ) νiziF
(31)
We have employed the superscript T for the transposed matrix. The phenomenological coefficient matrices L and λ(j) of
i
ti κ λiψ ) νiziF
(30)
Consequently
ti κ D + 2 2 2 ∂µs/∂cs ν z F i
( ) 1 ν1 z F 1 ν2 0
( )
2
λii )
(29)
1 0 M(2) ) ν1 z1F z2F
(15)
1 j ) -λ22∇µs - λ2ψ∇ψ1 ν2 2
λψψ ) κ
( ) 1 ν2 z1F z2F 0
λψψ )
2
zizjlijF2 ∑ ∑ i)1 j)1
(38)
where lij (i, j ) 1, 2) are the customary phenomenological coefficients.10,12 Thus these λij (i, j ) 1, 2, ψ) can be calculated either from tabulated lij by eqs 36-38 or directly from experimental data by eqs 18-20. A related formalism was presented by Førland et al. several years ago.1-4 They work with the same thermodynamic forces, but the fluxes involved are different: they use electrolyte fluxes, and we work with ion constituent fluxes. We feel that when i * 0, it is not possible to define electrolyte fluxes. Frequently the phenomenological equations are desired in terms of concentration gradients. From eqs 13-17, we deduce
j ) 1, 2
(23)
i * j ) 1, 2
(24)
tiκ 1 ∇ψ j ) -Di(j)∇cs νi i νiziF j
i, j ) 1, 2
(39)
X ) (∇µ˜ 1, ∇µ˜ 2)
(25)
i ) -A(j)∇cs - κ∇ψj
j ) 1, 2
(40)
J ) (j1, j2)
(26)
( ) 1 j, i νi i
where Di(j) and A(j) are given by
Nonequilibrium Binary Electrolyte Solutions
Di(j) ) D +
titkκ
∂µs 2 ∂c νν zz F s
i, j ) 1, 2
J. Phys. Chem. B, Vol. 101, No. 29, 1997 5743
(41)
TABLE 1: Proposals for Defining Three Electric Potentials Oi by Postulating the Ion Activity Coefficients γi γi
∇µi
φ
γ1 ) 1 γ2 ) 1 γ1 ) γ2 ) γs
∇µ1 ) RT∇ ln cs ∇µ2 ) RT∇ ln cs 1 ∇µ1 ) ∇µ2 ) ∇µs ν
φ1 φ2 φm
i k i k
A(j) )
tiκ ∂µs νiziF ∂cs
i * j ) 1, 2
(42)
In eq 41, if i * j, we take k ) i; and if i ) j, we take k * i. From eqs 39 and 40 we obtain the “superposition” result
ti 1 i ji ) -D∇cs + νi νiziF
i ) 1, 2
(43)
which is also obtained in the customary formalism.12 Pure diffusion occurs when i ) 0. In this case the electrolyte diffuses as a whole, and the electrolyte flux will be js ) ji/νi (i ) 1, 2). Other Formalisms There are other formalisms which describe nonequilibrium electrolyte solutions with other electric potentials. Two ways have been developed to define the electric potential: (i) by splitting into two terms the electrochemical potential µ˜ i ) µi + ziFφ (i ) 1, 2), where µi is the chemical potential and (ii) by applying the Nernst-Planck-Poisson (NPP) formalism. In the first way, the definition must be completed by postulating the Nernst potential of the electrodes ∆µi/ziF. In Table 1 we give the three most common proposals; these postulate the ion activity coefficient in different ways. Thus, we obtain three different electric potentials φi (i ) 1, 2, m), characterized by the subscript i. These electric potential gradients are related to the OEP gradients by
RT ∇ ln cs ziF
i ) 1, 2
1 ∇µ ∇φm ) ∇ψi νziF s
i ) 1, 2
∇φi ) ∇ψi -
(44)
1 j ) -Di*∇cs - ziuiFcs∇ψ* νi i
i ) 1, 2
i ) -A*∇cs - κ∇ψ*
(52)
(53) (54)
where
(45)
i * j ) 1, 2
(55)
i ) 1, 2
(56)
i ) 1, 2
(57)
tiκ νicszi2F2
From eqs 53 and 54 we deduce
ti 1 i ji ) -D*∇cs + νi νiziF
(46) where the diffusion coefficient D* will be
j ) 1, 2, m
(47)
where δi(j) and R(j) are given by
δi(m) ) Di(i) +
i ) 1, 2
where ji* are the fluxes in the apparatus-fixed reference frame; Di* are the diffusion coefficients of the ionic species; ui are the ionic mobilities; ψ* is the NPP electric potential in the solution; and v is the bulk velocity. Here ci (i ) 1, 2) denote the ionic concentrations. The Nernst-Einstein equation Di* ) RTui will be assumed. When the link with the NPP formalism is considered, it is necessary to assume the following: (i) the electrolyte is completely dissociated (then ci (i ) 1, 2) also expresses the ion constituent concentration); (ii) the ion constituent fluxes are ji* - civ ≈ ji;15 and (iii) when ∇cs ) 0 the NPP phenomenological equations will be identical to the phenomenological equations of other formalisms. Then the NPP phenomenological equations will be
ui )
i ) 1, 2; j ) 1, 2, m
i ) -R(j)∇cs - κ∇φj
δi(j) ) Di(j) +
ji* ) -Di*∇ci - ziuiFci∇ψ* + civ
A* ) νizi(Di* - Dj*)F
where the subscript m refers to the mean activity coefficient of the electrolyte γs. Using eqs 39 and 40 and 44-45, we deduce the following phenomenological equations
tiκ 1 ∇φ ji ) -δi(j)∇cs νi νiziF j
We now consider the NPP formalism. Here the fluxes are expressed by13
i, j ) 1, 2
(48)
∂µs ννizi F ∂cs
i ) 1, 2
(49)
RTκ zjcsF
j ) 1, 2
(50)
κ ∂µs νzjF ∂cs
j ) 1, 2
(51)
νizizjcsF2 tiκ
R(m) ) A(j) +
(58)
MgCl2 + H2O Solutions
RTtiκ
2 2
R(j) ) A(j) +
D* ) D1*t2 + D2*t1
The φi (i ) 1, 2) are also the quasi-electrostatic potentials which were studied by Newman.13,14
We now apply the different formalisms described above to the system MgCl2 + H2O. When working electrodes are excluded, the electric current density will be effectively zero at every point, i.e., i ) 0. Under these conditions we deduce from eq 40 that, in a nonequilibrium binary solution, the difference in the value of the OEP between two points depends exclusively on their solute concentrations, and not on the concentration profile between them, as has been well confirmed.16 (For two or more solutes, the OEP does depend on the concentration profiles.5) Therefore functions of the kind ψi ) ψi(cs) (i ) 1, 2) and φi ) φi(cs) (i ) 1, 2, m) can be deduced. This is done below for MgCl2 + H2O solutions at 25 °C. The subscripts 1 and 2 denote the ion constituents Mg and Cl, respectively. In Figure 1 we show these functions from 0.092 to 4.9 mol dm-3. The ψ2 ) ψ2(cs) are the emf values given in Table II by Phang and Stokes17 for concentration cells with transference.
5744 J. Phys. Chem. B, Vol. 101, No. 29, 1997
Garrido et al.
Figure 1. Electric potentials in solutions from 0.092 to 4.9 mol dm-3 when i ) 0. Values of ψ2 from Phang and Stokes.17
Figure 3. Diffusion coefficients D and D* of the electrolyte. Data from Miller et al.20
Figure 2. Electric potentials in solutions from 0.0001 to 0.092 mol dm-3 when i ) 0. Data from Miller et al.20 (s s) ψ*; (‚‚‚) φm.
Figure 4. Coefficients Di(j) (i, j ) 1, 2) of eqs 40. Data from Miller et al.20
We assign a reference value ψ2 ) 0 at cs ) 0.092 mol dm-3. To deduce ψ1 ) ψ1(cs), we use eq 8 cast in terms of activity coefficients
dψ1 ) dψ2 +
νRT d ln msγs(m) ν1z1F
(59)
The ln msγs(m) values given by Rard and Miller18 are transformed from molality ms quantities to molarity cs quantities.19,20 The functions φi ) φi(cs) (i ) 1, 2, m) are deduced from eqs 44 and 45. Six electric potentials from 0.0001 to 0.0920 mol dm-3 are shown in Figure 2. To deduce ψ2 ) ψ2(cs), we use eq 40, i.e.,
dψ2 ) -
(
)
(c)
d ln γs νRT t 1+ ν1z1csF 1 d ln cs
dcs
(60)
and as we are working with dilute solutions we can assume that
(
1+
)(
)
d ln γs(c) d ln γs(m) ≈ 1+ d ln cs d ln ms
(61)
The data given in Table V of Miller et al.20 are used. Then we obtain ψ1 ) ψ1(cs) and φi ) φi(cs) (i ) 1, 2, m) as before. We also include in Figure 2 the electric potential ψ* used in the NPP formalism, which we obtain from eq 54, i.e.,
dψ* )
( )
t1 RT t2 dcs Fcs 2
(62)
Figure 5. Coefficients δi(j) (i ) 1, 2; j ) 1, 2, m) of eqs 47. Data from Miller et al.20 (s s) δ2(m); (s ‚ s) δ1(m).
Figures 1 and 2 show important differences between the electric potential-concentration functions. When cs f 0, all the electric potentials go to (∞. Thus the electric potential gradients in very dilute solutions can easily reach very high values. Fortunately this is compensated by the extremely small values of the specific electric conductance κ. From the results in Figure 2 we can deduce the corresponding liquid junction potential (ljp) between two solutions cs′ and cs′′. Usually this quantity is given by the difference in φi (i ) 1, 2, m) or ψ*. According to the electric potential chosen, we obtain very different values. Figures 3-7 show the phenomenological coefficients for dilute solutions obtained from the data in Table V of Miller et al.20 We also assume the approximation of eq 61. We emphasize that Figure 4 shows that Di(i) , Di(j) (i * j ) 1, 2); therefore, the terms Di(j)∇cs in eqs 39 have very different values
Nonequilibrium Binary Electrolyte Solutions
Figure 6. Coefficients A(j) (j ) 1, 2) and A* of eqs 41 and 55. Data from Miller et al.20
Figure 7. Coefficients R(j) (j ) 1, 2, m) of eqs 48. Data from Miller et al.20 The values of A* of eq 55 have also been included.
J. Phys. Chem. B, Vol. 101, No. 29, 1997 5745
Figure 9. Coefficients λ1ψ/N (a), λ2ψ/N (b), and l12/N (c). Data from Miller et al.20
Figure 10. Coefficients l22/N (a) and λψψ/N (b). Data from Miller et al.20
taken from Table VI of Miller et al.20 and the λij/N obtained from them using eqs 36 and 37. Division of lij and λij by N reduces significantly their change with concentration. Discussion
Figure 8. Coefficients λ22/N (a) and l11/N ) λ11/N (b). Data from Miller et al.20 N is the normality in equiv dm-3.
according to i * j or i ) j (i, j ) 1, 2). We see also that for highly dilute solutions Di(i) f 0 (i ) 1, 2). In Figure 5 we see that δi(j) f δi (i ) 1, 2; j ) 1, 2, m) when cs f 0; this quantity is expressed by
δi ) D +
RTtiκ (νti 2 2 νi zi csF2
- νk)
(i * k ) 1, 2) (63)
Figure 7 shows that all R(j) f A* (j ) 1, 2, m) when cs f 0. Figures 8-10 show the coefficients lij/N (i, j ) 1, 2) and λij/N (i, j ) 1, 2, ψ) for the solution MgCl2 + H2O at 25 °C up to 5.0 mol dm-3. N is the normality in equiv dm-3. The values of the ionic transport coefficients lij/N (i, j ) 1, 2) have been
The phenomenological equations proposed above consider a linear dependence between the forces and fluxes, as is usually accepted when solution transport processes are studied. In particular, they apply very well in electrolyte solutions for large values of the forces. As shown above, the experimental values of the phenomenological coefficients can be evaluated from experimental values of the classical coefficients ti, κ, and D. The new formalism uses the concept of ion constituents. Since it is independent of the degree of dissociation of the electrolyte, it can be applied to both strong or weak electrolytes at all values of solute concentrations, just as can the customary formalism.10,12,13 There are differences in the number of independent coefficients of the various formalisms. The new and customary formalisms compute their phenomenological coefficients from the three experimental transport quantities ti, κ, and D, together with the solute activity coefficient, an equilibrium property. Consequently they are both exact and complete. In contrast the NPP equations evaluate their coefficients from only the two transport quantities ti and κ and consequently are an incomplete and inexact approximation. More specifically, the NPP equations are fundamentally a dilute solution approximation which omits the cation-anion cross-terms. A relationship between the electric potential and the electric charge density F ) FΣzici, such as Poisson’s equation, is not
5746 J. Phys. Chem. B, Vol. 101, No. 29, 1997 proposed in the new formalism. We noted above that the phenomenological equations have been deduced from the dissipation function and not from Maxwell’s laws. The ∇ψi (i ) 1, 2), ∇φi (i ) 1, 2, m), and ∇ψ* values depend on ∇cs and i through the phenomenological coefficients. The values that F can reach have little interest. However, we can evaluate F if we assume a value for F0 at the initial time t0 and then compute the change through dF ) -∇i dt.21-23 Finally we contrast this new λij formalism with the customary lij formalism. The new one has the advantage of being formulated from the start with measurable quantities and also of using the electric current density as one of the fluxes, but the disadvantage that the numerical values of the transport properties of the solution λij depend on whether the electrodes are reversible to ion 1 or to ion 2. The customary one has the advantage that the values of the solution transport properties lij are independent of any electrodes, but the disadvantage that the formulation begins with unmeasurable quantities. However, the description of any real experiment in the customary formalism always results in the combination of the unmeasurable properties to form measurable ones.10,12 We also note that these “unmeasurable quantities” are considered unmeasurable because currently there is no practical experimental way to obtain them. However, in principle they could be obtained experimentally if it were possible to measure the rise time of the potential in the 10-9 s just after the formation of the boundary in a concentration cell with transference.22 Acknowledgment. This work is part of project PB95-0018 of the DGICYT (Ministry of Education and Science of Spain). Portions of the work of D.G.M. were done under the auspices of the Office of Basic Energy Sciences (Geosciences) of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract No. W-7405-ENG-48. D.G.M. also
Garrido et al. thanks the DGICYT and the Departamento de Termodina´mica of the Universitat de Vale`ncia for their hospitality during his visit in May-June 1995. References and Notes (1) Førland, K. S.; Førland, T. J. Stat. Phys. 1995, 78, 513. (2) Førland, K. S.; Førland, T.; Ratkje, S. K. Acta Chem. Scand. A 1977, 31, 47. (3) Førland, K. S.; Førland, T.; Ratkje, S. K. AdVances in Thermodynamics; Taylor and Francis: New York, 1992; Vol. 6, p 340. (4) Førland, K. S.; Førland, T.; Ratkje, S. K. IrreVersible Thermodynamics: Theory and Applications; John Wiley: New York, 1988. (5) Garrido, J.; Compan˜, V.; Lo´pez, M. L. J. Phys. Chem. 1994, 98, 6003. (6) Garrido, J.; Mafe´, S.; Aguilella, V. M. Electrochim. Acta 1988, 33, 1151. (7) Garrido, J.; Compan˜, V.; Aguilella, V. M.; Mafe´, S. Electrochim. Acta 1990, 35, 705. (8) Garrido, J.; Compan˜, V. J. Phys. Chem. 1992, 96, 2721. (9) Garrido, J.; Compan˜, V.; Lo´pez, M. L. Electrochim. Acta 1993, 38, 877. (10) Haase, R. Thermodynamics of IrreVersible Processes; Dover: New York, 1990; p 293. (11) Guggenheim, E. A. Thermodynamics; North-Holland: Amsterdam, 1967; p 299. (12) Miller, D. G. J. Phys. Chem. 1966, 70, 2639. (13) Newman, J. Electrochemical Systems; Prentice-Hall: London, 1973; pp 76 and 217. (14) Smyrl, W. H.; Newman, J. J. Phys. Chem. 1968, 72, 4660. (15) We remember that v ) ΣViji*. When i ) 0 we know that v ) Vsjs* + Vwjw*, and we obtain ji* - civ ) cwVw ji ≈ ji. This also requires cs to be relatively small, corresponding to fairly dilute solutions. (16) Brown, A. S.; MacInnes, D. A. J. Am. Chem. Soc. 1935, 57, 1356. (17) Phang, S.; Stokes, R. H. J. Solution Chem. 1980, 9, 497. (18) Rard, J. A.; Miller, D. G. J. Chem. Eng. Data 1981, 26, 38. (19) Dunlop, P. J.; Gosting, L. J. J. Phys. Chem. 1959, 63, 86. (20) Miller, D. G.; Rard, J. A.; Eppstein, L. B.; Albright, J. G. J. Phys. Chem. 1984, 88, 5739. (21) Hafemann, D. R. J. Phys. Chem. 1965, 69, 4226. (22) Leckey, J. H.; Horne, F. H. J. Chem. Phys. 1981, 85, 2504. (23) Mafe´, S.; Pellicer, J.; Aguilella, V. M. J. Phys. Chem. 1986, 90, 6045.
